Evaluating Divergence Thm: V, S & Verify for x^2+y^2+z^2=1

In summary, the divergence theorem states that the triple integral over V of the divergence of V with respect to x, y, and z is equal to the double integral over S of the dot product of V and the normal vector with respect to the surface area. In this particular problem, evaluating both sides of the equality involves integrating the simple div V over the volume V and the more challenging V dot n over the surface S, which can be simplified using polar coordinates after projecting the surface onto the xy-plane. Both the part above and below the xy-plane must be considered.
  • #1
jlmac2001
75
0
I need help evaluating both sides of the divergence theorem if V=xi+yj+zk and the surface S is the sphere x^2+y^2+z^2=1, and so verify the divergence theorem for this case.

Is the divergence theorem the triple integral over V (div V) dxdydz= the double integral over S (V dot normal)dS? If so I would I evaluate it for the above problem?
 
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  • #2
Try looking these things up, (Wolfram/mathworld). And yes, verify means evaluate both sides of the equality.
 
  • #3
Note that, in this problem, div V is very simple so the integration over the volume is trivial. Integrating V.n dS on the surface is a bit more challenging but if you "project" the surface into the xy-plane and then use polar coordinates, it should be easy. (Don't forget to do both the part of the sphere above the xy-plane and the part below!)
 

1. What is the Divergence Theorem?

The Divergence Theorem is a mathematical concept that relates the surface integral of a vector field over a closed surface to the triple integral of the divergence of that vector field over the enclosed volume.

2. How is the Divergence Theorem used to evaluate a vector field?

The Divergence Theorem allows us to evaluate a vector field by converting a surface integral into a triple integral, which can often be easier to solve. This is particularly useful for complicated 3D shapes.

3. What is the significance of V and S in the Divergence Theorem?

V represents the enclosed volume, while S represents the closed surface that bounds that volume. These are important components of the Divergence Theorem as they allow us to relate the surface integral to the triple integral.

4. How do I verify the Divergence Theorem for a specific function or surface?

To verify the Divergence Theorem for a specific function or surface, you will need to calculate the surface integral and the triple integral separately and then compare the results. If they are equal, the Divergence Theorem holds for that function and surface.

5. Can the Divergence Theorem be applied to any vector field and surface?

No, the Divergence Theorem is only applicable to certain types of vector fields and surfaces. The vector field must be continuous and differentiable, while the surface must be smooth and closed. Additionally, the surface must be orientable, meaning that there is a consistent normal direction at every point on the surface.

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